Appendix A. Proofs (Informative)

This appendix contains proofs of theorems contained in
Section 5 of the document.

A.1 Correspondence between Abstract OWL and OWL DL

This section shows that the two semantics,
the direct model theory for
abstract OWL ontologies from
Section 3,
here called the direct model theory,
and the OWL DL semantics from
Section 5,
here called the OWL DL model theory,
correspond on certain OWL ontologies.

All URI references used as ontology names are taken from VA,
class IDs are taken from VC,
datatype IDs are taken from VD,
individual IDs are taken from VI,
individual-valued property IDs are taken from VOP,
data-valued property IDs are taken from VDP,
annotation property IDs are taken from VAP, and
ontology property IDs are taken from VXP.

All URI references used as individual IDs are given a type in some
ontology in O.

The theorem to be proved is then:
Let O and O' be collections of OWL DL ontologies in abstract syntax form
that are imports closed,
such that their union has a separated vocabulary.
Then
O direct entails O'
if and only if
T(O)
OWL DL entails
T(O')
the translation of the ontologies in O'
(exending T to allow it to translate collections of ontologies in the
obvious way).

To make the induction work, it is necessary to show that for any
d a description or data range with sub-constructs
T(d) contains triples for each of
the sub-constructs that do not share any blank nodes with triples from
the other sub-constructs.
This can easily be verified from the rules for T.

If p∈VOP then I satisfies p∈IOOP.
Then, as I is an OWL DL interpretation,
I satisfies
<p,I(owl:Thing)>∈EXTI(I(rdfs:domain))
and
<p,I(owl:Thing)>∈EXTI(I(rdfs:range)).
Thus I satisfies T(p).
Similarly for p∈VDP.

Base Case: v ∈ VC, including owl:Thing and owl:Nothing

As v∈VC and I satisfies T(V),
thus
I(v)∈CEXTI(I(owl:Class)).
Because I is an OWL DL interpretation
CEXTI(I(v))⊆IOT,
so <I(v),I(owl:Thing)>∈EXTI(I(rdfs:subClassOf)).
Thus I OWL DL satisfies T(v).
As M(T(v)) is v, thus CEXTI(I(M(T(v))))=EC(v).
Finally, from above, I(v)∈IOC.

Base Case: v ∈ VD, including rdfs:Literal

As v∈VD and I satisfies T(V),
thus
I(v)∈CEXTI(I(rdfs:Datatype)).
Because I is an OWL DL interpretation
CEXTI(I(v))⊆LVI, so
<I(v),I(rdfs:Literal)>∈EXTI(I(rdfs:subClassOf)).
Thus I RDF-compabile satisfies T(v).
As M(T(v)) is v, thus CEXTI(I(M(T(v))))=EC(v).
Finally, from above I(v)∈IDC.

Base Case: d=oneOf(i1…in),
where the ij are individual IDs

As ij∈VI for 1≤j≤n and I satisfies T(V),
thus I(ij)∈IOT.
The second comprehension principle for sequences then requires that
there is some l∈IL that is a sequence of
I(i1),…,I(in)
over IOT.
For any l that is a sequence of
I(i1),…,I(in)
over IOT
the comprehension principle for oneOf requires that there is some
y∈CEXTI(I(rdfs:Class)) such that
<y,l> ∈ EXTI(IS(owl:oneOf)).
From the third characterization of oneOf, y∈IOC.
Therefore I satisfies T(d).
For any I+A that satisfies T(d),
CEXTI(I+A(M(T(d)))) =
{I(i1),…,I(in)}
= EC(d).
Finally, I+A(M(T(d)))∈IOC.

Base Case: d=oneOf(v1…vn),
where the vi are data literals

Because I(vj)∈LVI,
the second comprehension principle for sequences then requires that
there is some l∈IL that is a sequence of
I(v1),…,I(vn)
over LVI.
For any l that is a sequence of
I(v1),…,I(vn)
over LVI
the comprehension principle for oneOf requires that there is some
y∈CEXTI(I(rdfs:Class)) such that
<y,l> ∈ EXTI(IS(owl:oneOf)).
From the second characterization of oneOf, y∈IOC.
Therefore I satisfies T(d).
For any I+A that satisfies T(d),
CEXTI(I+A(M(T(d)))) =
{I(i1),…,I(in)}
= EC(d).
Finally, I+A(M(T(d)))∈IDC.

Base Case: d=restriction(p value(i)),
with p∈VOP∪VDP and i an individualID

From the induction hypothesis, I satisfies T(d').
As d' is a description, from the induction hypothesis
there is a mapping, A, that maps all the blank
nodes in T(d') into domain elements such that I+A satisfies T(d')
and I+A(M(T(d'))) = EC(d') and I+A(M(T(d')))∈IOC.
The comprehension principle for complementOf then requires
that there is a y∈IOC such that I+A satisfies
<y,e>∈EXTI(I(owl:complementOf))
so I satisfies T(d).
For any I+A that satisfies T(d),
CEXTI(I+A(M(T(d)))) =
IOT-CEXTI(I+A(M(T(d)))) = R-EC(d') = EC(d).
Finally, I+A(M(T(d)))∈IOC.

Inductive Case: d = unionOf(d1 … dn)

From the induction hypothesis,
I satisfies di for 1≤i≤n so
there is a mapping, Ai, that maps all the blank
nodes in T(di) into domain elements such that
I+Ai satisfies T(di).
As the blank nodes in T(di) are disjoint from the blank
nodes of T(dj) for i≠j,
I+A1+…+An, and thus I, satisfies
T(di)∪…∪T(dn).
Each di is a description, so from the induction hypothesis,
I+A1+…+An(M(T(di)))∈IOC.
The first comprehension principle for sequences then requires that
there is some l∈IL that is a sequence of
I+A1+…+An(M(T(d1))),…,
I+A1+…+An(M(T(dn))) over IOC.
The comprehension principle for unionOf then requires
that there is some y∈IOC such that
<y,l>∈EXTI(I(owl:unionOf))
so I satisfies T(d).
For any I+A that satisfies T(d), I+A satisfies T(di) so
CEXTI(I+A(di)) = EC(di)).
Then CEXTI(I+A(M(T(d)))) =
CEXTI(I+A(d1))∪…∪CEXTI(I+A(dn))
= EC(d1)∪…∪EC(dn) = EC(d).
Finally, I(M(T(d)))∈IOC.

Inductive Case: d = intersectionOf(d1 … dn)

Similar.

Inductive Case: d = restriction(p x1 x2
… xn)

As p∈VOP∪VDP, from above I satisfies T(p).
From the induction hypothesis,
I satisfies restriction(p xi) for 1≤i≤n so
there is a mapping, Ai, that maps all the blank
nodes in T(restriction(p xi)) into domain elements such that
I+Ai satisfies T(restriction(p xi)).
As the blank nodes in T(restriction(p xi)) are disjoint from the blank
nodes of T(restriction(p xj)) for i≠j,
I+A1+…+An, and thus I, satisfies
T(restriction(p x1 … xn)).
Each restriction(p xi) is a description, so from the induction hypothesis,
M(T(restriction(p xi)))∈IOC.
The first comprehension principle for sequences then requires that
there is some l∈IL that is a sequence of
I+A1+…+An(M(T(restriction(p xi)))),…,
I+A1+…+An(M(T(restriction(p xi)))) over IOC.
The comprehension principle for intersectionOf then requires
that there is some y∈IOC such that
<y,l>∈EXTI(I(owl:intersectionOf))
so I satisfies T(d).
For any I+A that satisfies T(d) I+A satisfies T(di) so
CEXTI(I+A(di)) = EC(di)).
Then CEXTI(I+A(M(T(d)))) =
CEXTI(I+A(restriction(p xi)))∩…∩
CEXTI(I+A(restriction(p xn)))
= EC(restriction(p xi))cap;…∩EC(restriction(p xi))
= EC(d).
Finally, I(M(T(d)))∈IOC.

As p∈VOP∪VDP, from above I satisfies T(p).
From the induction hypothesis, I satisfies T(d').
As d' is a description, from the induction hypothesis,
any mapping, A, that maps all the blank
nodes in T(d') into domain elements such that
I+A satisfies T(d') has
I+A(M(T(d'))) = EC(d') and I+A(M(T(d')))∈IOC.
As p∈VOP∪VDP and I satisfies T(V'),
I(p)∈IOOP∪IODP.
The comprehension principle for allValuesFrom restrictions then requires
that I satisfies the triples in T(d) that are not in T(d') or T(p) in
a way that shows that I satisfies T(d).
For any I+A that satisfies T(d),
CEXTI(I+A(M(T(d)))) =
{x∈IOT |
∀ y∈IOT : <x,y>∈EXTI(p) implies
y∈CEXTI(M(T(d')))} =
{x∈R | ∀ y∈R : <x,y>∈ER(p) implies y∈EC(d')}
= EC(d).
Finally, I+A(M(T(d)))∈IOC.

Lemma 1.1:
Let V', V, I', and I be as in
Lemma 1.
Let d be an abstract OWL individual construct over V',
(of the form Individual(…)).
Then for any A mapping all the blank nodes of T(d) into RI
where I+A OWL DL satisfies T(d),
I+A(M(T(d))) ∈ EC(d).
Also, for any r ∈ EC(d) there is some
A mapping all the blank nodes of T(d) into RI
such that I+A(M(T(d))) = r.

Proof:

A simple inductive argument shows that
I+A(M(T(d))) must satisfy all the requirements of EC(d).
Another inductive argument, depending on the non-sharing of blank nodes in
sub-constructs, shows that for each r ∈ EC(d) there is some
A such that I+A(M(T(d))) = r.

A.1.2 Correspondence for Directives

Lemma 1.9:
Let V', V, I', and I be as in
Lemma 1.
Let F be an OWL directive over V' with an annotation of the form
annotation(p x).
If F is a class or property axiom, let
n be the name of the class or property.
If F is an individual axiom, let n be the main node of T(F).
Then for any A mapping all the blank nodes of T(F) into RI,
I+A OWL DL satisfies the triples resulting from the annotation
iff I' direct satisfies the conditions resulting from the annotation.

Proof:

For annotations to URI references, the lemma can be
easily established by an inspection of the semantic condition and the
translation triples.
For annotations to
Individual(…), the use
of Lemma 1.1 is also needed.

Lemma 2:
Let V', V, I', and I be as in
Lemma 1.
Let F be an OWL directive over V'.
Then I satisfies T(F) iff I' satisfies F.

Proof:

The main part of the proof is a structural induction over directives.
Annotations occur in many directives and work exactly the same so
they just require a use of
Lemma 1.9.
The rest of the proof will thus ignore annotations.
Deprecations can be handled in a simlar fashion and will also be ignored in
the rest of the proof.

Case: F = Class(foo complete d1 … dn).

Let d=intersectionOf(d1 … dn).
As d is a description over V', thus I satisfies T(d) and
for any A mapping the blank nodes of T(d)
such that I+A satisfies T(d), CEXTI(I+A(M(T(d)))) = EC(d).
Thus for any sub-description of d, d',
CEXTI(I+A(M(T(d')))) = EC(d'),
and I+A(M(T(d')))∈IOC.
Thus for some A mapping the blank nodes of T(d)
such that I+A satisfies T(d), CEXTI(I+A(M(T(d)))) = EC(d)
and I+A(M(T(d)))∈IOC;
and for each d' a sub-description of d,
CEXTI(I+A(M(T(d')))) = EC(d'),
and I+A(M(T(d')))∈IOC.

If I' satisfies F then EC(foo) = EC(d).
From above, there is some A such that
CEXTI(I+A(M(T(d)))) = EC(d) = EC(foo) =
CEXTI(I(foo))
and I+A(M(T(d)))∈IOC.
Because I satisfies T(V),
I(foo)∈IOC,
thus <I(foo),I+A(M(T(d)))> ∈
EXTI(I(owl:equivalentClass)).
Further, because of the semantic conditions on
I(owl:intersectionOf),
<I(foo),I+A(M(T(SEQ d1 … dn)))> ∈
EXTI(I(owl:intersectionOf)).

If d is of the form intersectionOf(d1)
then
CEXTI(I+A(M(T(d1)))) =
EC(d1) = EC(d) = EC(foo)
and I+A(M(T(d1)))∈IOC.
So, from the semantic conditions on
I(owl:equivalentClass),
<I(foo),I+A(M(T(d1)))> ∈
EXTI(I(owl:equivalentClass)).
If d1 is of the form complementOf(d'1)
then
IOT - CEXTI(I+A(M(T(d'1)))) =
CEXTI(I+A(M(T(d1)))) =
EC(d1) = EC(d) = EC(foo)
and I+A(M(T(d'1)))∈IOC.
So, from the semantic conditions on
I(owl:complementOf),
<I(foo),I+A(M(T(d'1)))> ∈
EXTI(I(owl:complementOf)).
If d1 is of the form unionOf(d11 … d1m)
then
CEXTI(I+A(M(T(d11)))) ∪ …
∪ CEXTI(I+A(M(T(d1m)))) =
CEXTI(I+A(M(T(d1)))) =
EC(d1) = EC(d) = EC(foo)
and I+A(M(T(d1j)))∈IOC, for 1≤ j ≤ m.
So, from the semantic conditions on
I(owl:unionOf),
<I(foo),I+A(M(T(SEQ d11 … d1m)))> ∈
EXTI(I(owl:unionOf)).

Therefore I satisfies T(F), for each potential T(F).

If I satisfies T(F)
then I satisfies T(intersectionOf(d1 … dn)).
Thus there is some A as above such that
<I(foo),I+A(M(T(d)))> ∈
EXTI(I(owl:equivalentClass)).
Thus EC(d) = CEXTI(I+A(M(T(d)))) =
CEXTI(I(foo)) = EC(foo).
Therefore I' satisfies F.

Case: F = Class(foo partial d1 … dn)

Let d=intersectionOf(d1 … dn).
As d is a description over V', thus I satisfies T(d) and
for any A mapping the blank nodes of T(d)
such that I+A satisfies T(d), CEXTI(I+A(M(T(d)))) = EC(d).
Thus CEXTI(I+A(M(T(di)))) = EC(di),
for 1 ≤ i ≤ n.
Thus for some A mapping the blank nodes of T(d)
such that I+A satisfies T(d),
CEXTI(I+A(M(T(di)))) = EC(di),
and I+A(M(T(di))∈IOC,
for 1 ≤ i ≤ n.

If I' satisfies F then EC(foo) ⊆ EC(di),
for 1 ≤ i ≤ n.
From above, there is some A such that
CEXTI(I+A(M(T(di)))) =
EC(di) ⊇
EC(foo) = CEXTI(I(foo))
and I+A(M(T(di))∈IOC.
Because I satisfies T(V),
I(foo)∈IOC,
thus <I(foo),I+A(M(T(di)))> ∈
EXTI(I(rdfs:subClassOf)),
for 1 ≤ i ≤ n.
Therefore I satisfies T(F).

If I satisfies T(F)
then I satisfies T(di),
for 1 ≤ i ≤ n.
Thus there is some A as above such that
<I(foo),I+A(M(T(di)))> ∈
EXTI(I(rdfs:subClassOf)),
for 1 ≤ i ≤ n.
Thus EC(d) = CEXTI(I+A(M(T(di)))) ⊇
CEXTI(I(foo)) = EC(foo),
for 1 ≤ i ≤ n.
Therefore I' satisfies F.

Case: F = EnumeratedClass(foo i1 … in)

Let d=oneOf(i1 … in).
As d is a description over V' so I satisfies T(d) and
for some A mapping the blank nodes of T(d)
such that I+A satisfies T(d),
EC(d) = CEXTI(I+A(M(T(d)))) =
{SI(M(T(i1)), … SI(M(T(in))}
Also, SI(M(T(ij)) ∈ IOT, for 1 ≤ j ≤ n.

If I' satisfies F then EC(foo) = EC(d).
From above, there is some A such that
CEXTI(I+A(M(T(d)))) = EC(d) = EC(foo) =
CEXTI(I(foo))
and I+A(M(T(d))∈IOC.
Let e be I+A(M(T(SEQ i1 … in))).
Then, from the semantic conditions on
I(owl:oneOf),
<I(foo),e> ∈
EXTI(I(owl:oneOf)).
Therefore I satisfies T(F).

The only thing that needs to be shown here is the typing for foo,
which is similar to that for classes.

Case: F= DisjointClasses(d1 … dn)

As di is a description over V' therefore I satisfies
T(di) and for any A mapping the blank nodes of T(di)
such that I+A satisfies T(di),
CEXTI(I+A(M(T(di)))) = EC(di).

If I satisfies T(F)
then for 1≤i≤n there is some Ai
such that I satisfies
<I+Ai(M(T(di))),I+Aj(M(T(dj)))>
∈ EXTI(I(owl:disjointWith))
for each 1≤i<j≤n.
Thus EC(di)∩EC(dj) = {}, for i≠j.
Therefore I' satisfies F.

If I' satisfies F
then EC(di)∩EC(dj) = {} for i≠j.
For any Ai and Aj as above
<I+Ai+Aj(M(T(di))),I+Ai+Aj(M(T(dj)))>
∈ EXTI(I(owl:disjointWith)), for i≠j.
As at least one Ai exists for each i,
and the blank nodes of the T(dj) are all disjoint,
I+A1+…+An satisfies
T(DisjointClasses(d1 … dn)).
Therefore I satisfies T(F).

As di for 1≤i≤m is a description over V' therefore I satisfies
T(di) and for any A mapping the blank nodes of T(di)
such that I+A satisfies T(di),
CEXTI(I+A(M(T(di)))) = EC(di).
Similarly for ri for 1≤i≤k.

If I' satisfies F, then, as p∈VOP,
I satisfies I(p)∈IOOP.
Then, as I is an OWL DL interpretation,
I satisfies
<I(p),I(owl:Thing)>∈EXTI(I(rdfs:domain))
and
<I(p),I(owl:Thing)>∈EXTI(I(rdfs:range)).
Also, ER(p)⊆ER(si) for 1≤i≤n, so
EXTI(I(p))=ER(p) ⊆
ER(si)=EXTI(I(si))
and I satisfies
<I(p),I(si)>∈EXTI(I(rdfs:subPropertyOf)).
Next, ER(p)⊆EC(di)×R for 1≤i≤m, so
<z,w>∈ER(p) implies z∈EC(di) and
for any A such that I+A satisfies T(di),
<z,w>∈EXTI(p) implies
z∈CEXTI(I+A(M(T(di))))
and thus
<I(p),I+A(M(T(di)))>∈EXTI(I(rdfs:domain)).
Similarly for ri for 1≤i≤k.

If I' satisfies F and inverse(i) is in F, then
ER(p) and ER(i) are converses.
Thus <u,v>∈ER(p) iff <v,u>∈ER(i)
so <u,v>∈EXTI(p) iff
<v,u>∈EXTI(i)
and I satisfies
<I(p),I(i)>∈EXTI(I(owl:inverseOf)).
If I' satisfies F and Symmetric is in F, then
ER(p) is symmetric.
Thus if <x,y>∈ ER(p) then <y,x>∈ER(p)
so if <x,y> ∈ EXTI(p)
then <y, x>∈EXTI(p).
and thus
I satisfies p∈CEXTI(I(owl:Symmetric)).
Similarly for Functional, InverseFunctional, and Transitive.
Thus if I' satisfies F then I satisfies T(F).

If I satisfies T(F) then, for 1≤i≤n,
<I(p),I(si)>∈EXTI(I(rdfs:subPropertyOf))
so ER(p)=EXTI(I(p)) ⊆
EXTI(I(si))=ER(si).
Also, for 1≤i≤m, for some A such that I+A satisfies T(di),
<I(p),I+A(M(T(di)))>∈EXTI(I(rdfs:domain))
so <z,w>∈EXTI(p) implies
z∈CEXTI(I+A(M(T(di)))).
Thus <z,w>∈ER(p) implies z∈EC(di) and
ER(p)⊆EC(di)×R.
Similarly for ri for 1≤i≤k.

If I satisfies T(F) and inverse(i) is in F, then
I satisfies
<I(p),I(i)>∈EXTI(I(owl:inverseOf)).
Thus <u,v>∈EXTI(p) iff
<v,u>∈EXTI(i)
so <u,v>∈ER(p) iff <v,u>∈ER(i)
and ER(p) and ER(i) are converses.
If I satisfies F and Symmetric is in F, then
I satisfies p∈CEXTI(I(owl:Symmetric))
so if <x,y> ∈ EXTI(p)
then <y, x>∈EXTI(p).
Thus if <x,y>∈ ER(p) then <y,x>∈ER(p)
and ER(p) is symmetric.
Similarly for Functional, InverseFunctional, and Transitive.
Thus if I satisfies T(F) then I' satisfies F.

As pi∈VOP and I satisfies T(V'),
I(pi)∈IOOP.
If I satisfies T(F) then
<I(pi),I(pj)>
∈ EXTI(I(owl:equivalentProperty)),
for each 1≤i<j≤n.
Therefore EXTI(pi) = EXTI(pj),
for each 1≤i<j≤n;
ER(pi) = ER(pj), for each 1≤i<j≤n;
and I' satisfies F.

If I' satisfies F
then ER(pi) = ER(pj), for each 1≤i<j≤n.
Therefore EXTI(pi) = EXTI(pj),
for each 1≤i<j≤n.
From the OWL DL definition of owl:equivalentProperty,
<I(pi),I(pj)>
∈ EXTI(I(owl:equivalentProperty)),
for each 1≤i<j≤n.
Thus I satisfies T(F).

If I satisfies T(F)
then there is some A that maps each blank node in T(F)
such that I+A satisfies T(F).
A simple examination of T(F) shows that the mappings of A plus
the mappings for the individual IDs in F, which are all in IOT,
show that I' satisfies F.

If I' satisfies F
then for each Individual construct in F
there must be some element of R that makes the type relationships and
relationships true in F.
The triples in T(F) then fall into three categories.
1/ Type relationships to owl:Thing,
which are true in I because the elements above belong to R.
2/ Type relationships to OWL descriptions,
which are true in I because they are true in I',
from Lemma 1.
3/ OWL property relationships, which are true in I' because they
are true in I.
Thus I satisfies T(F).

V', V, I', and I meet the requirements of
Lemma 2,
so for any directive D over V'
I satisfies T(D) iff I' satisfies D.

Because O is imports closed,
O includes all the ontologies that would be imported in T(O)
the importing part of imports directives will be handled the same.
Satisfying an abstract ontology is just satisfying its directives
and satisfying the translation of an abstract ontology is just satisfying all
the triples so
I OWL DL satisfies T(K) iff I' direct satisfies K.

The extensions of
owl:allValuesFrom,
owl:cardinality,
owl:hasValue,
owl:maxCardinality,
owl:minCardinality,
owl:onProperty, and
owl:someValuesFrom
are as necessary to link the elements of IOC and IDC up with their parts.
Their class extensions are all empty.

The extensions of
owl:complementOf,
owl:intersectionOf,
owl:oneOf, and
owl:unionOf
are as necessary to make their
semantic conditions
work out correctly.
Their class extensions are all empty.
This can easily be done here because modifying these extensions does not
induce any loops in the process.

CEXTI(SI(owl:AllDifferent)) consists
of those elements of IAD that have an owl:distinctMembers property
EXTI(SI(owl:differentFrom))
is the inequality relation on U
EXTI(SI(owl:disjointWith))
relates members of IOC that have the disjoint class extensions
EXTI(SI(owl:distinctMembers))
relates elements of IAD to their copy in IL, but only for sequences of
distinct individuals
EXTI(SI(owl:equivalentClass))
relates members of IOC that have the same class extension
EXTI(SI(owl:equivalentProperty))
relates members of IOP∪IDP that have the same extension
EXTI(SI(owl:inverseOf))
relates members of IOP whose extensions are inverses of each other
EXTI(SI(owl:sameAs))
is the equality relation on U
EXTI(SI(owl:AllDifferent)) =
CEXTI(SI(owl:differentFrom)) =
CEXTI(SI(owl:disjointWith)) =
CEXTI(SI(owl:distinctMembers)) =
CEXTI(SI(owl:equivalentClass)) =
CEXTI(SI(owl:equivalentProperty)) =
CEXTI(SI(owl:inverseOf)) =
CEXTI(SI(owl:sameAs)) =
{}

Then I is an OWL DL interpretation because the conditions for the class
extensions in OWL DL match up with the conditions for class-like OWL
abstract syntax constructs.

V', V, I', and I meet the requirements of
Lemma 2,
so for any directive D over V'
I satisfies T(D) iff I' satisfies D.

Because O is imports closed,
O includes all the ontologies that would be imported in T(O)
the importing part of imports directives will be handled the same.
Satisfying an abstract ontology is just satisfying its directives
and satisfying the translation of an abstract ontology is just satisfying all
the triples so
I OWL DL satisfies T(K) iff I' direct satisfies K.

A.1.5 Correspondence Theorem

Theorem 1:
Let O and O' be collections of OWL DL ontologies in abstract syntax form
that are imports closed,
such that their union has a separated vocabulary, V',
and every URI reference in V' is used in O.
Then
O entails O'
if and only if
T(O)
OWL DL entails
T(O')
the translation of the ontologies in O'
(exending T to allow it to translate collections of ontologies in the
obvious way).

Then I satisfies T(V'), because each URI reference in V' is used on O.

Proof:
Suppose O entails O'.
Let I be an OWL DL interpretation that satisfies T(O).
Then from Lemma 3,
there is some direct interpretation I' such that
for any abstract OWL ontology X over V',
I satisfies T(X) iff I' satisfies X.
Thus I' satisfies each ontology in O.
Because O entails O', I' satisfies O',
so I satisfies T(O').
Thus T(K),T(V') OWL DL entails T(Q).

A.2 Correspondence between OWL DL and OWL Full

This section contains a proof sketch concerning the relationship between
OWL DL and OWL Full.
This proof has not been fully worked out. Significant effort may be
required to finish the proof and some details of the relationship may have
to change.

Let K be an RDF graph.
An OWL interpretation of K is an OWL interpretation
(from Section 5.2)
that is an D-interpretation of K.

Lemma 5:
Let V be a separated vocabulary.
Then for every OWL intepretation I there is an OWL DL interpretation
I' (as in Section 5.3)
such that for K any OWL ontology in the abstract
syntax with separated vocabulary V,
I is an OWL interpretation of T(K) iff I' is an OWL DL interpretation of T(K).

Proof sketch:
As all OWL DL interpretations are OWL interpretations, the reverse
direction is obvious.

Let I = < RI, EXTI, SI, LI >
be an OWL interpretation that satisfies T(K).
Let I' = < RI', EXTI', SI', LI' >
be an OWL interpretation that satisfies T(K).
Let RI' = CEXTI(I(owl:Thing)) + CEXTI(I(owl:ObjectProperty)) +
CEXTI(I(owl:ObjectProperty)) +
CEXTI(I(owl:Class)) + CEXTI(I(rdf:List)) + RI,
where + is disjoint union.
Define EXTI' so as to separate the various roles of the copies.
Define SI' so as to map vocabulary into the appropriate copy.
This works because K has a separated vocabulary, so I can be split
according the the roles, and there are no inappropriate relationships in
EXTI.
In essence the first component of RI' is OWL individuals,
the second component of RI' is OWL datatype properties,
the third component of RI' is OWL individual-valued properties,
the fourth component of RI' is OWL classes,
the fifth component of RI' is RDF lists,
and the sixth component of RI' is everything else.

Theorem 2:
Let O and O' be collections of OWL DL ontologies in abstract syntax form
that are imports closed,
such that their union has a
separated vocabulary
(Section 4.2).
Then
the translation of the ontologies in O
OWL Full entails
the translation of the ontologies in O'
if
the translation of the ontologies in O
OWL DL entails
the translation of the ontologies in O'.

Proof:
From the
above lemma
and because all OWL Full interpretations are OWL
interpretations.